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ISBN 978-952-62-0836-7 (Paperback)ISBN 978-952-62-0837-4 (PDF)ISSN 0355-3213 (Print)ISSN 1796-2226 (Online)

U N I V E R S I TAT I S O U L U E N S I SACTAC

TECHNICA

U N I V E R S I TAT I S O U L U E N S I SACTAC

TECHNICA

OULU 2015

C 532

Mikko Leinonen

FINITE ELEMENT METHOD AND EQUIVALENT CIRCUIT BASED DESIGN OF PIEZOELECTRIC ACTUATORS AND ENERGY HARVESTER DYNAMICS

UNIVERSITY OF OULU GRADUATE SCHOOL;UNIVERSITY OF OULU,FACULTY OF INFORMATION TECHNOLOGY AND ELECTRICAL ENGINEERING,DEPARTMENT OF ELECTRICAL ENGINEERING

C 532

ACTA

Mikko Leinonen

C532etukansi.kesken.fm Page 1 Tuesday, May 12, 2015 12:55 PM

A C T A U N I V E R S I T A T I S O U L U E N S I SC Te c h n i c a 5 3 2

MIKKO LEINONEN

FINITE ELEMENT METHOD AND EQUIVALENT CIRCUIT BASED DESIGN OF PIEZOELECTRIC ACTUATORS AND ENERGY HARVESTER DYNAMICS

Academic dissertation to be presented with the assent ofthe Doctoral Training Committee of Technology andNatural Sciences of the University of Oulu for publicdefence in the OP auditorium (L10), Linnanmaa, on 26June 2015, at 12 noon

UNIVERSITY OF OULU, OULU 2015

Copyright © 2015Acta Univ. Oul. C 532, 2015

Supervised byDocent Jari JuutiProfessor Heli Jantunen

Reviewed byProfessor Stephen BeebyProfessor Pasi Kallio

ISBN 978-952-62-0836-7 (Paperback)ISBN 978-952-62-0837-4 (PDF)

ISSN 0355-3213 (Printed)ISSN 1796-2226 (Online)

Cover DesignRaimo Ahonen

JUVENES PRINTTAMPERE 2015

Leinonen, Mikko, Finite element method and equivalent circuit based design ofpiezoelectric actuators and energy harvester dynamics University of Oulu Graduate School; University of Oulu, Faculty of Information Technologyand Electrical Engineering, Department of Electrical EngineeringActa Univ. Oul. C 532, 2015University of Oulu, P.O. Box 8000, FI-90014 University of Oulu, Finland

Abstract

The main objective of this thesis was to use and combine Finite Element Method (FEM) and smallsignal equivalent circuit models in actuator and energy harvesting design and to study thedynamics of the said designs.

The work is divided into four different sections. In the first section, the small signal parametersare derived for a pre-stressed piezoelectric actuator using a series of measurements. In addition,the tunability of the resonance frequency using mass and series capacitors is investigated.

In the second section, a piezoelectric Fabry Perot Interferometer actuator is simulated usingFEM and the small signal parameters are derived using FEM simulations. The modelled resultsare compared with the actual measurements and the resonance frequency is found to differ by only0.8 percent from the measured values when the mirror is attached to the actuator.

In the third section a piezoelectric wide band energy harvester is developed with multiple beamtopology. Two different designs are compared, one produced using the conventional PZT-steelstructure and one with a PZT-LTCC structure.

The final section presents an FEM model for a shoe mounted energy harvester and concentrateson the modelling of walking dynamics in FEM. The simulation results are compared to actualmeasurements and the simulated power values are found to differ by only 7% when the cymbalstroke is below 1.3 mm. The generated model is also expandable to other types of energyharvesters and the methods developed can be used in a variety of different energy harvestingsimulations and harvester development.

The results show that the equivalent circuit approach together with FEM modelling is apowerful tool in the dynamics design of piezoelectric actuators and energy harvesters.

Keywords: actuator, energy harvester, FEM, piezoelectric

Leinonen, Mikko, Elementtimenetelmän ja vastinpiirien käyttö pietsosähköistenaktuaattorien ja energiankorjuukomponenttien dynamiikan suunnittelussaOulun yliopiston tutkijakoulu; Oulun yliopisto, Tieto- ja sähkötekniikan tiedekunta,Sähkötekniikan osastoActa Univ. Oul. C 532, 2015Oulun yliopisto, PL 8000, 90014 Oulun yliopisto

Tiivistelmä

Väitöstyön päätavoitteena oli yhdistää elementtimenetelmät (FEM) ja piensignaalimallit aktuaat-torien ja energiankorjuukomponenttien suunnittelussa ja tutkia niiden dynamiikkaa.

Työ on jaettu neljään eri osaan. Ensimmäisessä osassa piensignaalimallit johdettiin pietsosäh-köisestä aktuaattorista mittausten avulla. Lisäksi resonanssitaajuuden muuttamista tutkitaan mas-san ja sarjaan kytketyn kapasitanssin avulla.

Toisessa osassa simuloidaan pietsosähköistä Fabry Perot interferometria käyttäen elementti-menetelmää. Lisäksi komponentin piensignaalimalli luodaan käyttäen simulointimallia. Lopuksipiensignaalimallin ja prototyypin mittaustuloksia verrataan. Mallin resonanssitaajuus poikkeaamitatusta vain 0.8 %, kun aktuaattoriin on kiinnitetty peili.

Kolmannessa osassa kehitetään ja verrataan toisiinsa kahta erilaista laajakaistaista monipalk-kista pietsosähköistä energian korjuukomponenttia. Toinen komponenteista on toteutettu perin-teisellä PZT-teräs rakenteella ja toinen yhteissintratulla PZT-LTCC rakenteella.

Viimeisessä osassa luodaan simulaatio malli kenkään asennetulle cymbal tyyppiselle pietso-sähköiselle energian korjuukomponentille ja kävelyn dynamiikkaa tutkitaan. Luotua mallia ver-rataan prototyypin mittaustuloksiin ja simuloitu energian tuotto poikkeaa vain 7 % alle 1.3 mmpuristusliikkeellä.

Tulokset osoittivat, että piensignaalimallin ja elementtimenetelmän yhdistäminen on tehokasapu pietsosähköisten aktuaattorien ja energiankorjuukomponenttien dynamiikan suunnittelussa.

Asiasanat: aktuaattori, energiankorjuukomponentti, FEM, pietsosähköisyys

7

Acknowledgements

First of all, I wish to thank my supervisors Docent Jari Juuti and Professor Heli

Jantunen for supervising and supporting my work. Furthermore, I would like to

thank the colleagues at Microelectronics and Materials Physics laboratories for a

good working environment. Jaakko Palosaari, Maciej Sobocinski and Jari Juuti

have been of tremendous help in my work and for that I am grateful. Professor

A.E. Hill has also been very helpful in correcting my grammar in this thesis and

in our papers.

The Finnish Foundation for Technology Promotion, Wihuri Foundation,

KAUTE Foundation and GETA graduate school are acknowledged for financially

supporting this work.

Oulu, May 2015 Mikko Leinonen

8

9

List of abbreviations and symbols

α Overlapping factor

ε Permittivity

ξ Damping ratio ν Poisson’s ratio

ω Angular frequency AFM Atomic Force Microscope BW Bandwidth C Capacitance Cs Series capacitance CST Constant Strain Triangle CTE Coefficient of Thermal Expansion d Displacement

e error

E Young’s modulus

F Force FEM Finite Element Method FPI Fabry-Pérot Interferometer

fres Resonance frequency

GA Genetic Algorithm

IR InfraRed

k Spring constant keff Effective spring constant l Length of a beam LASER Light Amplification by Stimulated Emission of Radiation LCR Inductor, capacitor and resistor circuit LTCC Low Temperature Co-fired Ceramic LVDT Linear Variable Differential Transformer m Mass madded Added mass meff Effective mass MFC Macro Fiber Composite n Transformer turn ratio

NIR Near InfraRed

PZT Lead Zirconium Titanate Rin Input resistance

10

rm Mechanical resistive impedance representing mechanical losses

SOI Silicon On Insulator

SPICE Simulation Program with Integrated Circuit Emphasis

Vin Input voltage Vp-p Peak-to-peak voltage

x Displacement

11

List of original papers

This thesis is based on the following five original papers, which are cited in the

text by the given roman numerals.

I Leinonen M, Juuti J & Jantunen H (2005) Equivalent circuit based modification of a pre-stressed piezo actuator. Proc. 4th Int. Conf. on Smart Systems, Seinäjoki.

II Sobocinski M, Leinonen M, Juuti J & Jantunen H (2012) A Piezoelectric Active Mirror Suspension System Embedded Into Low-Temperature Cofired Ceramic. IEEE trans. on Ultrasonics, Ferroelectrics and frequency control 59(9): 1990-1995.

III Leinonen M, Palosaari J, Juuti J & Jantunen H (2011) Piezoelectric energy harvester for vibrating environments using multiple beam topology for wideband operation. AP XIX Proceedings 41, Helsinki.

IV Sobocinski M, Leinonen M, Juuti J & Jantunen H (2011) Monomorph piezoelectric wideband energy harvester integrated into LTCC. Journal Of European Ceramic Society 31(5): 789-794.

V Leinonen M, Palosaari J, Juuti J & Jantunen H (2014) Combined electrical and electromechanical simulations of a piezoelectric Cymbal harvester for energy harvesting from walking. Journal of Intelligent Material Systems and Structures, 25(4): 391-400.

In Paper I an equivalent circuit for a piezoelectric actuator is developed using

measurements and is then used to tune the resonance frequency of the actuator. In

Paper II a mirror suspension actuator is developed on LTCC for IR spectroscopy.

In this paper the FEM simulations are used as the main tool in the design. In

Paper III a wideband energy harvester is developed with 5 differently tuned

piezoelectric beams using a conventional PZT-Steel structure. In Paper IV an

improved wideband energy harvester is developed with LTCC technology in order

to improve the resonance frequency accuracy. In Paper V an energy harvester for

walking is modelled and simulated using FEM software and the results are

compared with actual measurements.

In Paper I the idea, measurements and the main part of the writing was done

by the author. In Paper II the idea, design and the modelling was done by the

author and the manufacturing and measurements were done by co-authors. In

paper III the idea, theory and design of the harvester as well as part of the analysis

of the results was

done by the author. The manufacturing and the measurements were done by the

co-authors. In Paper IV the idea of the energy harvester and the design as well as

the main part of the writing was done by the author. The manufacturing and the

measurements were done with the help of co-authors. In Paper V the development

12

of the simulation model, simulations and the main part of the writing were done

by the author and the idea and design of the harvester as well as the

measurements were done by the co-authors.

13

Contents

Abstract

Tiivistelmä

Acknowledgements 7

List of abbreviations and symbols 9

List of original papers 11

Contents 13

1 Introduction 15

1.1 Actuator and energy harvesting dynamics .............................................. 15

1.2 Scope and outline of the thesis ................................................................ 18

2 Small signal model parametrization for pre-stressed piezoelectric

actuator 19

2.1 The Butterworth-Van Dyke equivalent circuit ........................................ 19

2.2 Experimental development of the equivalent circuit parameters ............ 21

2.2.1 The design and manufacturing of the test device ......................... 21

2.2.2 Measurement results and the development of the

equivalent circuit model ............................................................... 22

3 FPI Actuator design 27

3.1 FPI mirror actuator .................................................................................. 27

3.2 The design of the FPI-actuator ................................................................ 27

3.3 The simulation model of the FPI-actuator ............................................... 28

3.4 The simulation results of the FPI-actuator .............................................. 30

3.5 The equivalent circuit parameters of the FPI-actuator ............................ 31

3.6 Verification of the equivalent circuit parameters of the FPI-

actuator using measurements .................................................................. 32

4 Energy harvester designs 35

4.1 Wide band energy harvester .................................................................... 35

4.1.1 PZT-steel wideband energy harvester .......................................... 39

4.1.2 LTCC wideband energy harvester ................................................ 40

4.1.3 Measurement results and comparison between the

harvesters ...................................................................................... 41

4.2 Modelling energy harvesters for walking ................................................ 47

4.2.1 The design of the shoe mounted energy harvester ........................ 48

4.2.2 The FEM model of the energy harvester ...................................... 49

4.2.3 The simulation results of the energy harvester ............................. 52

5 Conclusions 59

14

References 63

Original papers 71

15

1 Introduction

Piezoelectric actuators, sensors and energy harvesters are widely used in different

applications throughout industry and are also encountered in many everyday tasks

in people’s lives. These piezoelectric components can be categorized as

transducers that either transform electrical energy into kinetic energy (actuators)

or vice versa (energy harvesters and sensors). The rate of this energy

transformation is defined by the mode of actuation, sensing or energy production.

These modes can be roughly divided into two operating groups as static and

dynamic operation modes. Static or quasistatic operation is encountered in

actuation and sensing but usually not in energy harvesting. Dynamic operation, on

the other hand, is used in all of the transducer applications. The dynamics of the

transducers play an important role in the design of the said components. For

example, the actuators and sensors in micropositioning systems should preferably

work outside resonance in order to maintain a linear frequency response and

avoid ringing. Energy harvesting applications, on the other hand, usually benefit

greatly from operating in resonance. Some actuator applications such as

ultrasonic actuators are also dependent on operation at resonance frequencies.

1.1 Actuator and energy harvesting dynamics

Piezoelectric actuators come in many sizes and can be used to drive bulky

equipment as well as micro- or even nano-scale objects. Applications where large

objects are actuated include, for example, telescopes such as in the Japanese

Subaru telescope where the mirror is actively suspended, thus avoiding the

disadvantages of a rigid mirror [1,2]. Similarly, a telescopic active hexapod

utilizes piezoelectric actuators [2,3]. In [4] a method was presented to replace the

conventional control surfaces with piezoelectric composite fibres (MFC)

embedded into an aircraft wing. A similar concept was presented in [5] with

adaptive control implemented using piezoelectric actuators. Piezoelectric

actuators are, however, more prominent at the smaller end of the size spectrum.

The atomic force microscope (AFM) and scanning probe microscope both utilize

piezoelectric actuators in the manipulation of the tip [2,6]. The resolution of these

devices is in the nanometre scale and with small scanning areas (5x5x1 µm) the

resolution can reach 0.1 nm in the XY-direction and 0.015 nm in the Z-direction

[7,8]. The nonlinear nature of the piezoelectric actuators means that, in

micropositioning, control systems are needed together with the sensors to

16

linearize the system. Closed loop control, however, requires that the dynamical

properties of the actuator are known in order to guarantee stability. Some open

loop control schemes are also viable; for example, current drive, where the effect

of the nonlinear capacitance of the piezoelectric component is avoided by using

charge as the input. This reduces hysteresis, which can be around 15 % without

any control [9], and requires no instrumentation other than metering of the current

[9,10]. For example, in [11], the initial hysteresis of 27 % was reduced to only

2 % by using current drive. In current drive the model for the piezoelectric

component is not needed, but can be included to improve performance [11].

However, in the case of both open (feed-forward) and closed loop control the

model needs to be known. The exception to this is the use of various adaptive

control systems, where the parameters of the model are not known a priori, but

rather generated iteratively during the operation of the device [12].

In some open loop control methods the system is modelled and then the

inverse of the model is applied as the controller, thus altering the dynamics and

nonlinearities of the system. This method relies on very accurate modelling of the

system and in some cases only the nonlinearities are modelled while preserving

the dynamical characteristics of the system [2,13]. The hysteresis can be modelled

with various methods such as the Preisach model and its variant Prandtl-Ishlinskii

models. These models have yielded good results in hysteresis cancellation: for

example an uncontrolled error of 17.5 % was reduced to 2 % using the Prandtl-

Ishlinskii model in [14].

Closed loop control, by its very nature, requires a feedback signal from the

output. For example, in micropositioning applications this signal derives from the

displacement of the actuator and in acoustic devices this can be the sound

pressure signal from a microphone or hydrophone. The feedback control requires

a high quality feedback signal because any interference and noise added to the

feedback signal is also treated as an input signal and is therefore usually passed

unattenuated to the output. In micropositioning applications the movements are

small, usually in the range of a few micrometres or even nanometres, and

measuring such small movements makes high demands on the instrumentation of

the system. Methods commonly used in conjunction with piezoelectric actuators

include Linear Variable Differential Transformer (LVDT), Light Amplification by

Stimulated Emission of Radiation (LASER), capacitive, strain gauge and

piezoresistive sensors [15-19]. In addition, the model of the system should be

known in order to control the system reliably, as is the case with most of the

control systems described.

17

The Finite Element Method (FEM) is traditionally used in mechanical

engineering to study the strain and stress in solid material. FEM is therefore well

suited for modelling piezoelectrical components and various examples of design

based on FEM modelling exist, for example in fuel injector design [20,21]. In

energy harvesting FEM modelling can be used to design structures where

analytical methods can be too difficult to realize. FEM modelling can also enable

optimization with methods such as genetic algorithms (GA) [22].

Small signal models have been traditionally used in electric circuit design,

but mechanical systems can be modelled using the same circuit components as in

electric circuit models. Piezoelectric components, having both electrical and

mechanical components, are therefore well suited for equivalent circuit analysis

where lumped circuit parameters represent the mass, spring and losses of the

component. The most classical equivalent circuit is the Butterworth Van Dyke

circuit [23]. This model is usable when the first resonance mode of the component

is under examination and the operation is considered linear. Nonlinearities such as

impact can be modelled, for example, by adding nonlinear components to the

circuit [24]. In this work, the small signal model is used to study the dynamics of

the actuators and to tune the resonance frequency. Nonlinear parameters are

encountered in the studied cymbal harvester, and can be used to explain the

nonlinear behaviour under high load conditions.

Harvesting energy from vibration is most efficiently done at the resonance

frequency of the piezoelectric component, and when designing a wideband energy

harvester the resonances have to be designed in such a way as to create a pass-

band. This task also requires a knowledge of the dynamics of the vibrating

piezoelectric components. This can be acquired by creating a small signal model

of the vibrating component. In particular, in multiple beam wideband energy

harvesters, the individual beams’ resonance frequencies have to be tuned in order

to create a wide pass-band. This task requires a knowledge of the beam’s

dynamics and this can be acquired by creating a small signal model of the beam.

With the aid of this model the dynamics can then be designed in such a way that

the required wideband operation is achieved. Multiple beam harvesters have been

manufactured using electromagnetic [25] and piezoelectric beams [26].

Piezoelectric bimorphs with varying piezoelectric layer thicknesses were used in

[27] to achieve the desired resonance frequency distribution. Beam length was

varied in [28] and masses were varied in [28,29].

18

1.2 Scope and outline of the thesis

In this work, small signal models are generated for a multitude of actuators and

energy harvesters. The emphasis is on the resonance frequency behaviour and

therefore the Butterworth-Van Dyke equivalent circuit is used as the model. The

parameters were obtained either by measurements or by FEM-simulations.

In Chapter 2, an equivalent circuit model is obtained for a pre-stressed

piezoelectric component. The pre-stressing makes it difficult to obtain the

parameters using analytical methods alone and thus they were obtained using a

series of measurements.

In Chapter 3, a Fabry-Pérot Interferometer (FPI) actuator is developed and

characterized. The dynamics of the mirror actuator are critical in interferometer

applications and therefore the dynamical model is required. The model was

obtained using FEM and then compared using measurements.

In Chapter 4, two different wideband energy harvesters are designed. The

principle of operation is dependent on the accurate control of resonance

frequencies in both the design and manufacturing phases. Actuators were

manufactured by two different methods and the measurement results were

compared to pinpoint design and manufacturing aspects which affect the

dynamical behaviour of the wideband energy harvester. In addition, a model for

an energy harvester which harvests energy from walking is developed. The

developed model is shown to accurately model the energy output from the

piezoelectric energy harvester when it is installed into the sole of a shoe. Also, the

dynamics of the energy harvester are examined, as the spring constant in this case

is highly nonlinear and the implications of this are further discussed.

The purpose of this work is to provide tools for small signal model creation

and show the importance of these models in actuator and energy harvester design.

Actuator dynamics can be tuned and evaluated before manufacturing and

previously manufactured components can be characterized and their small signal

model can be created using methods presented in this work. In energy harvesting,

better wideband energy harvesters can be manufactured by using small signal

models. The concept of spring constant can also be used in studying the nonlinear

behaviour encountered in high displacement shoe mounted piezoelectric energy

harvesters.

19

2 Small signal model parametrization for pre-stressed piezoelectric actuator

2.1 The Butterworth-Van Dyke equivalent circuit

Pre-stressed piezoelectric actuators have not been properly modelled or

parameterized before. Multilayer pre-stressed piezoelectric stack actuator was

modelled as a simple 2nd degree system and parameterized using step response in

[30], but bending pre-stressed actuators have not been properly parameterized.

The pre-stressing changes both the mechanical and electrical properties of the

actuator [31] and therefore parametrization should be done using measurements.

The most commonly used equivalent circuit is the Butterworth-Van Dyke circuit

[32] shown in Fig. 1. Other more complicated equivalent circuits have also been

developed [33-35], but the classical Butterworth-Van Dyke is the simplest and

easiest to implement. It consists of a voltage source Vin, the voltage source’s

output resistance Rin, the capacitance C of the piezo component, a transformer

with a turn ratio of n, and the mechanical losses which are modelled as rm. The

effective mass of the component is modelled as an inductor with an inductance of

m and the spring constant, k, is shown in this circuit as a capacitor. The electrical

domain in the left side of the circuit has only the capacitance of the piezo

component and the mechanical domain on the right side of the circuit is

comprised of an inductor, capacitor and resistor (LCR) circuit which effectively

determines the resonance frequency of the component. This is true when the

coupling between the electrical domain and the mechanical domain is weak i.e.

the transformer ratio is low, which is usually the case in piezoelectric devices.

20

Fig. 1. The classical Butterworth-Van Dyke equivalent circuit for piezoelectric

components [Paper I, published by permission of Frami OY].

The resonance frequency, when assuming lossless component i.e. rm=0, for the

circuit in Fig. 1. can be expressed as

(1)

and the transformer ratio by

(2)

where F is the force output of the component under Vin excitation. As can be seen,

the most obvious method to tune the resonance frequency is to alter the mass of

the component by adding mass, madded, to the point of the greatest displacement

under vibration. Then the effective mass becomes meff = m+madded .

When current drive is used, for example to reduce the hysteresis of the

component, the capacitance of the piezoelectric component couples to the

resonating LCR-circuit and the resulting resonance frequency is

(3)

As can be seen, the capacitance stiffens the component by adding to the spring

constant. Thus the new effective spring constant is

(4)

As the spring constant can be altered by the capacitance of the piezoelectric

component, the capacitance should be tunable. This, however, still requires the

21

current drive for the component and in some cases this is not viable. The same

capacitance tuning can be obtained with voltage drive by using the circuit shown

in Fig. 2. Now a capacitance Cs is connected in series with the actuator and in

effect prevents the voltage source from grounding the capacitance of the

piezoelectric component.

Fig. 2. The equivalent circuit for a piezoelectric actuator with a series capacitor

[Paper I, published by permission of Frami OY].

The equation for the resonance frequency is now

(5)

From this equation, it can be seen that the resonance frequency can be tuned

between the voltage driven case in equation (1) and current driven case in

equation (3). For example, if Cs is 0, the resonance frequency is given by (3) and

when Cs >>C, then resonance frequency is obtained by equation (1).

2.2 Experimental development of the equivalent circuit parameters

2.2.1 The design and manufacturing of the test device

The equivalent circuit was developed for a simple pre-stressed unimorph bender,

which was clamped from both ends as shown in Fig. 3. The dimensions of the

piezoelectric element were 5 x 34 x 0.25 mm. The material was PZT-5H from

Vin

1:n

C

m

1/k

CS

22

Morgan Electro Ceramics Inc. The piezoelectric disc, 35 mm in diameter, was

pre-stressed by printing 60 µm Ag-paste on one side and then firing at 850 °C.

The coefficient of thermal expansion (CTE) mismatch between the paste and the

PZT material created the pre-stress. The details of this process are presented in

[36] and [37]. The pre-stressed disc was then cut with a Nd:YVO4 laser and poled

in a 6 MV/m electric field at 100 °C for 30 min. A steel strip with dimensions of 5

x 38 x 0.05 mm was glued on the PZT beam and then fixed to clamps 0.7 mm

from the edges of the PZT beam. A mirror was attached to the centre of the beam

for subsequent displacement measurements using a Michelson interferometer

[38].

Fig. 3. The pre-stressed unimorph piezoelectric disc clamped from both sides [Paper I,

published by permission of Frami OY].

2.2.2 Measurement results and the development of the equivalent

circuit model

The measurements consisted of displacement and force measurements. The

displacement measurements were done with a Michelson interferometer detailed

in [38] and with a signal generator, Agilent 33120A. The force measurements

were performed using a Kyowa WGA-650B instrumentation amplifier and a

Kyowa LUB-20KB force sensor.

The transformer ratio, n, was obtained using force measurements. The

actuator was loaded with 1 N mechanical pre-stress by a force sensor and then

actuated with a varying voltage and the change in the force reading recorded. Due

to the low sensitivity of the force sensor, relatively high voltages had to be used.

Using equation (2) the n together with the error estimate was plotted in Fig. 4.

The error estimate was based on the ±0.01 N error of the force reading. The

transformer ratio, n, at 200 V was recorded as 2.8 mN/V [Paper I].

23

Fig. 4. The transformer ratio n and the error estimate versus the input voltage Vin

[Paper I, published by permission of Frami OY].

A displacement measurement was used to determine the spring constant, k. Under

static conditions k=nVin/x and by simply measuring the displacement x under Vin

excitation using the previously obtained n the k was calculated as 15.6 kN/m

[Paper I].

A resonance measurement was used together with equation (1) to determine

the effective mass, meff, which in this case is m, since madded was 0. The resonance

was obtained by using displacement measurement and sweeping the frequency.

The resulting resonance frequency was found to be 1053 Hz and, using equation

(1), the effective mass meff was obtained as 0.296 g [Paper I].

The capacitance of the piezoelectric component was measured to be 30 nF

and if the mechanical losses are omitted i.e. rm=0, then the equivalent circuit is

fully parameterized for studying the effects of added mass and an external

capacitor on the resonance frequency. The effect of mass was studied by adding

masses i.e. madded in 0.3 g increments, with starting point of 0.779 g i.e. the mass

of the bolt.. The resonance frequencies at each mass are shown in Fig. 5 where the

calculated values using the small signal parameters and equation (1) are also

plotted. As can be seen, the calculated values differed only slightly from the

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0 20 40 60 80 100 120 140 160 180 200

n

Voltage (V)

nError

24

measured values. The greatest relative error between the calculated and the

measured value was 2.5 % at 1.08 g of added mass, as seen in Table 1 [Paper I].

Fig. 5. The resonance frequency versus the added mass of the actuator [Paper I,

published by permission of Frami OY].

Table 1. The errors between the measured and the estimated resonance frequencies

[Paper I, published by permission of Frami OY].

added mass (g) 0.779 1.08 1.38 1.68 1.98

error (%) 0.6 2.5 1.4 0.3 -1.9

The effect of external capacitance with a simple voltage drive was studied by

measuring the resonance frequencies of the prototype and varying the external

capacitance, Cs, in the circuit shown in Fig. 2. Four different capacitors were used

in the measurements with capacitances 10, 22, 47 and 100 nF respectively and the

change in the resonance frequency is shown in Fig. 6 where the analytical results

are also shown using equation (5). The prototype had a madded of 0.779 g in the

measurements. The error of the frequency measurement was ±0.5 Hz, since the

minimum step size for signal generator was 1 Hz, and is shown in Fig. 6 as

vertical bars [Paper I]. The measured values closely followed the analytical values

400

450

500

550

600

650

2.01.91.71.51.31.10.90.7

Freq

uenc

y (H

z)

Mass (g)

Estimated resonance freq.Measured resonance freq.

25

and therefore, taking into account the added mass measurements, the small signal

equivalent circuit parameters were successfully obtained.

Fig. 6. The change in resonance frequency of the prototype versus the series

capacitance [Paper I, published by permission of Frami OY].

A pre-stressed piezoelectric component was manufactured and its small signal

model was characterized. When comparing measured and calculated resonance

frequencies between the component and its model, a good agreement was found.

The model, however, did not incorporate losses, which could therefore affect the

obtained simulated results. Nevertheless, as the results show, the absolute error

between the component and its model was 2.5 % at maximum and the omission of

rm seems to be a valid decision. The methods provided to parametrize the model

can be used in a variety of piezoelectric actuators. Furthermore, the frequency

tuning using the model is a useful tool for tuning vibrating piezoelectric

components. Series capacitance modulation, i.e. using a tunable capacitor in

series with the component, can be used to fine tune the resonance of the

component. Energy harvesting applications can benefit from this approach since

the frequency can be tuned during operation and thus maintain the optimal

harvesting frequency when using high Q-factor harvesters. A high Q-value in

energy harvesters is beneficial in terms of obtained energy, but also means that

26

the energy can only be harvested from a very narrow frequency band. Therefore

tunability of components in energy harvesting is an ongoing research endeavor.

Mechanical pre-stress devices have been proposed [39], also electrostatic springs

with a comb structure [40] and a piezoelectric method [41] where the vibrating

beam is bent by a secondary beam to change the shape of the vibrating beam.

Electrical tuning methods have been proposed, for example, by adding a shunt

capacitance [42] which is similar to the method proposed in this work. The

difference is that even with low load impedances the series capacitance can shift

the frequency and therefore provide frequency tuning by shielding the electrical

side of the small signal model from grounding. This was demonstrated using an

inherently low impedance voltage drive on an actuator and recording the change

of resonance frequency as shown in Fig. 6. The tunability of the component is

naturally dependent of n, as is shown in [43].

27

3 FPI Actuator design

3.1 FPI mirror actuator

In spectroscopy a Fabry-Perot-interferometer (FPI) may be used to filter desired

wavelengths from infrared spectra to determine, for example, the chemical

composition of a medicine. Tunability of this interferometer is desirable because

the filtered spectrum can then be swept and analyzed. An FPI-filter usually

consists of two partially reflecting mirrors with an air gap between the mirrors.

The air gap determines the filter frequency and in order to tune this frequency a

straightforward approach is to move one or both of the mirrors. Various actuating

methods have been used previously such as piezoelectric stack actuators [44-46],

thermal actuators [47] and electrostatic actuators [48]. The near infrared channel

(NIR) is between 700 – 1000 nm and therefore piezoelectric actuation is well

suited for the FPI-filter since the displacement range is ~1 µm which can easily

be achieved with piezoelectric materials. The state of the art FPI-filter, of which

the author is one of the inventors [46], is shown in [49, 50] where stack-type

piezo actuators are installed in the recesses on the side of the mirrors. The

drawbacks of the design are the cost of the actuators and the mechanical stresses

they introduce to the mirror, which therefore has to be thick in order to provide

stiffness to the structure. In this work, the stack actuators are replaced with piezo

beams, which results in reduced costs and allows the use of smaller and lighter

mirrors. The bending beams, however, can also induce some stresses to the mirror

and therefore flexural joints are introduced to enable the use of smaller and

thinner mirrors in the filter. The dynamical properties of the FPI-actuator are then

examined using both the FEM-method and the small signal method. The aim of

the work was to provide a more cost effective alternative to the state of the art

actuator with smaller and lighter mirrors.

3.2 The design of the FPI-actuator

The goal of this work was to develop an FPI-filter in a TO-3 package, which

meant that the size of the filter was constrained to a diameter of 15 mm with a 3

mm optical aperture. TO-3 package was set as a goal, since it is a standard

package used in optical measurement instruments. A three actuator structure was

used to provide mirror tilting properties for the filter. This enabled the zeroing of

28

the manufacturing-introduced optical gap variation, i.e. the mirror tilt. Fig. 7.

shows the layout of the filter. The actuators were implemented as beams, as

opposed to the stack actuators used previously [44-46]. This reduced the required

voltage and provided high displacements in a relatively compact size. The

material of the beams was PZ-29 from Ferroperm and the passive layer was

Heraeus Heralock 2000 zero shrinkage LTCC tape. This manufacturing process is

described in [51-52]. The beams were connected to the mirror holder using a

flexural joint. Flexural joints are frequently used in micropositioning stages [53-

55] and their design and theory can be obtained from [56-57]. Flexural joints were

used to minimize the stress induced in the mirror because stress can bend the

mirror and therefore degrade the optical performance of the filter. In cases where

the mirror bends, the optical gap between the mirrors varies and therefore the

effective aperture changes. The hinges also reduce the stiffness of the beams and

therefore increase the displacement of the mirror.

Fig. 7. The FPI-mirror actuator layout.

3.3 The simulation model of the FPI-actuator

The purpose of the modelling of the FPI-actuator was to minimize the stress

induced in the partially reflecting mirror attached to the mirror holder. Also, the

dynamics of the actuator were of interest because the settling time of a gap

adjustment determines the frequency sweep duration. In addition to the simulation

of the actuator dynamics, static steady state simulations were carried out in order

29

to determine the obtainable gap modulation i.e. the displacement of the mirror and

the stresses induced in the mirror and the flexural joints.

Comsol Multiphysics software (Comsol Inc., Burlington, MA) was used in

the FEM modeling. The model for the FPI-actuator was developed by modelling

only the beams and the mirror holder because the supporting rim structure was

assumed to be rigid. Therefore the beams were only modelled to have a fixed

constraint at the base, since the actuator was glued with cyanoacrylate onto a

platform. The active layer was modelled as PZ-29 with the material parameters

shown in Table 2, obtained from [58]. The passive layer was modelled as

isotropic material with a Young’s modulus E of 85 GPa, Poisson’s ratio ν of 0.24

and density of 2900 kg/m3 obtained from [59]. The material properties of the

mirror were obtained from [60] with E of 74 GPa and ν of 0.17. The meshed

geometry is shown in Fig. 8.

Table 2. The material properties of PZ-29, PZT-5H, steel and plastic cushions, [38,53]

[Paper V].

Material density SE11,22 SE

12,21 SE13,23,31,32 SE

33 SE44,55 SE

66 d31,32 d33 d15,24 ε11,12 ε33 E ν

kg/m3 1x10-12 m2/N 1x10-12 m2/N GPa

PZ-29 7460 17 -5.78 -8.79 22.9 54.1 45.6 -243 574 724 2440 2870

PZT-5H 7500 16.5 -4.78 -8.45 20.7 43.5 42.6 -274 593 741 3130 3400

Steel 7850 200 0.33

Plastic cushion 1150 2 0.4

Fig. 8. The meshed geometry of the FPI-actuator [Paper II, published by permission of

IEEE].

30

3.4 The simulation results of the FPI-actuator

The first simulation was a static displacement model where a potential of 100 V

was applied to all of the beams. The displacement was recorded from the edge of

the mirror holder. The resulting displacement was 1.87 µm. However, the actual

measured value was ~1 µm, using OFV-5000 vibrometer (Polytech GmbH,

Germany). This error could be explained partly by the poling of the actual

component but the bending of the structure could also be a factor since the

simulation model omitted the supporting rim structure completely. Some

deformation can occur at the base of the beams and was also not included in the

model. The same simulation was used to study the stresses induced in the hinges

under actuation. The von Mises stress levels were recorded as 37.2 MPa, as

shown in Fig. 9. This is still well below the fracture limit of 200 MPa for the

LTCC material used [58].

Fig. 9. The von Mises stress (MPa) in the hinge.

The dynamic simulations were done using the eigenfrequency analysis mode on

the Comsol Multiphysics software. The first resonance mode was located at 14.7

kHz, whereas the measured resonance was 13.02 kHz. Other resonance modes

that were simulated are shown in Fig. 10 [Paper II]. As can be seen, the first

resonance is the first mode of a simple beam and the second and the third

resonance modes are the tilting resonance modes of the whole structure. The

fourth resonance mode is the second bending mode of a beam and it does not

include the tilting of the mirror. Analysing these results, it can be deduced that the

operating frequencies for this actuator are well below the first resonance of 14.7

31

kHz. However, some tilting of the mirror was observed at the 21.2 kHz resonance

and this has to be taken into account when driving the actuator in continuous

operation because mirror tilt could degrade the FPI-operation.

Fig. 10. The first resonance, second, third and fourth resonances at a) 14.7 kHz

b) 21.2 kHz c) 71.9 kHz and d) 77.4 kHz respectively [Paper II, published by permission

of IEEE].

3.5 The equivalent circuit parameters of the FPI-actuator

The dynamical properties of the FPI-actuator were studied with a mechanical

equivalent circuit similar to Fig. 1. In this case, however, only the resonance

behaviour was studied and therefore the equivalent circuit was reduced to

incorporate only the inertial mass, m, and the spring constant, k. The resonance

frequency of this system is stated in equation (1). The total effective mass of the

system was m=meff+mm, where meff is the effective mass of the moving parts of

FPI-module and mm is the mass of the attached mirror. The parameters for

equivalent circuit were obtained by simulating the structure with varying mirror

weights. For example if we have two different mirrors with weights m1 and m2,

the effective mass can be obtained as

32

(6)

where f1 is the resonance frequency with mass m1 and f2 is the resonance

frequency with mass m2. The spring constant can be then obtained as

2 (7)

The structure was simulated using a parametric sweep for the mirror density, in

order to change the mirror mass. The simulation was carried out as an

eigenfrequency simulation to obtain the resonance frequencies. Then using

Equation (6), the mean effective mass meff was obtained as 35.6 mg, using mirror

masses ranging from 14 mg to 61 mg. [Paper II]. Now, with the effective mass,

meff , using equation (7), the spring constant, k, was obtained as 326 kN/m [Paper

II]. With these values it is now possible to assess the dynamical performance of

the FPI-actuator using different sizes and with mirrors of different materials.

3.6 Verification of the equivalent circuit parameters of the FPI-

actuator using measurements

The resonance frequency of the prototype was measured both with and without a

mirror attached. The attached mirror had a diameter of 4 mm and a thickness of

0.7 mm. The mass of the mirror was 15 mg and thus the density was calculated as

1704 kg/m3. The measured resonance frequency without the mirror was 13.02

kHz and with the mirror it was 11.31 kHz. Thus the shift in resonance frequency

was 13.1 %. The calculated resonance frequencies using meff, k, and Equation (7),

were 14.7 kHz and 12.5 kHz without and with the mirror, respectively. The

change in resonance frequency was therefore 15.5 %. As can be seen, the absolute

values between the calculated and measured values differed by 12.9 % between

the mirror-less cases and by 10.5 % when the mirror was attached. The measured

change in resonance frequency after attaching the mirror was 1.71 kHz and the

simulated change was 2.2 kHz and 2.4 % in percentage points. However, when

the model is applied to the measured case, i.e. using the meff of 35.6 mg obtained

from simulations and calculating the new k using Equation (7), the new k is

obtained as 339 kN/m. With this value we can calculate the new resonance

frequency with the mirror as 11.4 kHz. The error is only 90 Hz or 0.8 % when

compared to the measured value of 11.31 kHz. The reason for the difference

between the simulated and measured results can be attributed to slight variations

33

in geometry and to the fact that the material parameters of both the passive layer

and the active layer change when PZT material is sintered together with LTCC

[31].

The results demonstrate that by using the equivalent circuit in conjunction

with FEM-modelling it is possible to characterize the linearized model, and to

use the model as a quick tool for estimating the contribution of the mirror’s

weight to the resonance frequency of the component. The procedure to do this is

first to use FEM-simulations to obtain an average meff and then calculate the k

using the measured resonance frequency of the FPI-actuator without the mirror,

and finally, using Equation (7), obtain the resonance frequency with the mirror

attached.

An FPI-actuator was developed using beam type actuators instead of stack

actuators, which are commonly used in FPI filters. The beam structure was

proven to provide the required displacement of 1.87 µm in simulation and a

measured value of 1 µm. This was obtained by using flexural hinges to reduce the

spring constant and also to minimize the stresses induced in the mirror. The

structure differs from state of the art structures by providing a more cost efficient

and lighter alternative while having a similar performance. A simple spring mass

model was parameterized using FEM-simulations to find the effective mass of the

actuator. The model then may be used as a quick tool for estimating frequency

shifts when the mass of the mirror is changed. The model also enables the

estimation of settling time using standard second order system equations, which is

also an important tool in actuator design.

34

35

4 Energy harvester designs

In many applications a power source is needed to provide energy for the sensors,

data processing and communication. The usual power sources are batteries,

accumulators and wired power lines. However, batteries need exchanging,

accumulators need recharging and power lines are costly to build and not so

convenient in many cases. To tackle these problems various energy harvesting

methods have been developed. Energy harvesting can be defined as collecting

small amounts of energy from the environment surrounding the component in

question. Energy sources typically include solar [61-62], vibration [63], pressure

[64-66], magnetic and RF-fields [67-69], chemical [70-71] and temperature [72-

74]. Piezoelectric materials are well suited for energy harvesting from vibration

and motion in general. Furthermore, the size of piezoelectric harvesters is usually

very compact and they are relatively easy to manufacture compared to

electrostatic and electromagnetic components.

4.1 Wide band energy harvester

In industrial environments vibration is usually encountered in machinery with

rotating parts. This provides a natural source of energy for sensors. When dealing

with rotary machinery, however, the majority of the vibration is present at the

angular frequencies at which the machines are rotating. This means that the

energy harvesters have to be tuned to the same frequency. Piezoelectrical

components have a high mechanical Q-value and therefore tuning the resonance

frequency can be difficult. For example, to match the rotational frequency of a

machine, the narrow bandwidth of the harvester requires a precise match to the

machine’s frequency. A wideband solution where the Q-value is reduced permits

frequency tuning which is much less critical. Also, if the machine has some

variation in the rotation speed, the wideband energy harvester is able to continue

harvesting the energy provided the frequency of the machine stays within the

bandwidth of the harvester. By the same argument, if the source of the vibration is

random, a wideband energy harvester is able to harvest more energy compared to

a conventional harvester.

Multiple beam wideband piezoelectric energy harvesters include [26,75-76]

and also an electromagnetic harvester in [25]. MEMS implementations include a

4 beam SOI structure [77] and a 2 beam micromachined structure [78]. Other

methods have been utilized such as coupled resonance structures [79]. Most

36

recent wideband harvesters use bistable operation, either with magnets [80],

buckled beams [81] or springs [82]. These energy harvesters are nonlinear as

opposed to multiple beam resonators which are usually linear in behaviour. The

multiple beam topology was chosen, because it can provide higher bandwidths by

increasing the number of beams. The drawback of this design is its sensitivity to

manufacturing tolerances, as will be shown. A method to incorporate

manufacturing tolerances into design parameters will be presented in the work.

The beam lengths are calculated using a simple recursion instead of the analytical

methods presented in [83-84]. The harvesters presented here were manufactured

using laser processing as in [26]. A PZT-Steel harvester consisted of a PZT layer

glued to a steel layer and a novel PZT-LTCC harvester consisted of a PZT layer

and an LTCC layer co-sintered together and then laser cut into shape in order to

minimize dimensional errors. This novel technique has not been previously

proposed in energy harvesters. The two harvesters are compared with the effect of

manufacturing tolerances in mind. The design of the wide band energy harvester

Two different wideband energy harvesters were designed and manufactured. A

multiple beam topology was chosen to accomplish the wideband operation. In this

beam topology the energy harvester comprises of multiple piezoelectric beams,

each with a different resonance frequency

In multiple beam topology, the individual resonance frequency of the n:th

beam is given as

, (8)

As the length of the beam is increased, the spring constant, kn , decreases and the

mass, mn , increases. The absolute values for the spring constant and mass are not

needed if only the changes in resonance frequency are required rather than the

actual resonance frequencies. Simple relations for the masses and spring constants

can be used to accomplish this resonance frequency tailoring. It is known from

beam theory that the spring constant of a beam has the relation

∝ (9)

, where l is the length of the beam. The mass of the beam has the relation

∝ (10)

Substituting these into equation (8), the relation for the resonance frequency is

∝ (11)

37

Now the difference in resonance frequencies between two beams using the beams’

damping ratio, ξ, and overlapping factor, α, is generated in this work as

, , , (12)

If the overlapping factor is 2, then the relative difference between the beams is

exactly the same as the relative bandwidth of the beam, namely 2ξ , which

coincides with the half power point in the frequency response. If an energy

harvester with beams separated with overlapping factor 2 is made, a flat power

spectrum is generated assuming that the power generated by each beam is

superimposed. When the overlapping factor is 0, the resonance frequencies are all

equal and the total bandwidth of the harvester is the same as that of a single beam,

i.e. 2ξ. The power, however, is the total sum of individual beams’ power. The

ideal value for the overlapping factor is therefore 2 with respect to wide band

operation and when using n beams the total bandwidth is 2ξn. A more general

expression for the bandwidth BW, with the inclusion of the overlapping factor, α,

is

2 1 (13)

Now, in order to design the wideband harvester, a recursion based on equations

(11) and (12) is developed as

, (14)

With this equation it is possible to design the resonance frequencies of the beams

to satisfy the bandwidth requirements. The starting frequency has to be either

calculated or simulated using FEM software to obtain the desired centre

frequency. It is also possible to alter the resonance frequency by adding mass to

the beams, but in order to taper the resonance frequencies, different masses are

required for individual beams.

During manufacturing, different errors are introduced into the dimensions of

the beams. The thickness, width and length of the beams can be altered by these

errors. For example, the error in beam length can be analyzed as follows. Using

equation (12) we get

,

,1 (15)

And by using equation (11), we get

1 (16)

38

now if absolute error e is introduced into ln we get

,

,~ 1 (17)

,where e/ln is the relative error. This equation shows the effect of error on the

resonance frequency shift. To compensate for this we define new overlapping

factor αc which can be obtained by setting

1 1 1 (18)

and solving for αc

1 1 1 (19)

where α is the desired overlapping factor. For example, we may set α as 2 and ξ as

0.025 and assume a positive relative error of 0.01 i.e. the actual length of the

beam is 1 % longer than the desired length and therefore the resonance frequency

is lower than expected. The compensated overlapping factor is obtained as 2.84.

This means that the beams are shortened to compensate for the error and the

resulting frequency shift is the same as with an overlapping factor of 2 without

any error present. In real world cases the error’s sign is rarely known a priori and

therefore a worst case scenario can be analyzed. If the sign of the error is opposite

between beams, then 2e can be used as a value for the error. In this case, if the

overlapping factor is chosen as 2, the worst case scenario widens the gap between

the two resonances in such a way that the ripple exceeds the 3dB limit. For

example, if we have a 1 % error in the length of the beams, we can use -2 % as

the relative error (2 % shorter beam than anticipated) with an overlapping factor 2

and ξ again as 0.025. Using equation (19) we get the compensated overlapping

factor as 0.32, which differs greatly from the original overlapping factor of 2.

This demonstrates the high sensitivity of the frequency tuning of multibeam

energy harvesters.

In this case, only the error in the length of the beam was analyzed, but by

using beam equations, the errors for width and thickness can also be analysed

using similar methods to those presented here.

39

4.1.1 PZT-steel wideband energy harvester

A five beam energy harvester was designed and manufactured. A circular closed

shape was chosen in order to facilitate encapsulation of the component. The

outline of the harvester is shown in Fig. 11.

Fig. 11. a) The dimensions and b) the manufactured prototype [Paper III, published by

permission of Suomen Automaatioseura].

The active material for the prototype was PZT-5H from Morgan Electroceramics.

The material parameters can be seen in [85]. The passive layer was steel and the

thicknesses of the active and passive layers were 375 µm and 150 µm

respectively. The length of the first beam was chosen as 22 mm, in order to fit as

long beams as possible inside the 35 mm piezo disc with sufficient support area.

The width of the beam was also maximised to 4 mm in order to improve the

power output of individual beams. It is important to maximize the beam length in

order to reduce the resonance frequency of the harvester because the natural

frequency is high for beams of this size. When the base frequency of the harvester

is made as low as possible, it is easier to reduce it further to the desired level by

adding mass at the tip of the beams. The design parameters for calculating the

length of the beams were 0.025 for the damping ratio, ξ, and 0.5 for the

overlapping factor, α. Damping ratio was chosen using a priori experience of

vibrating piezoelectric beams. A low value for the overlapping factor was chosen

in order to ensure proper overlapping and therefore higher power output at the

expense of the bandwidth, since hand gluing and placement generate higher

tolerances. These tolerances could be estimated as 100 µm for placement, but

glue thickness was not estimated, but could have been measured using a series of

40

cross-sectional cuts. The resulting beam lengths were 22, 21.9, 21.7, 21.6 and

21.4 mm.

4.1.2 LTCC wideband energy harvester

In addition to the traditional PZT-steel unimorph wideband energy harvester a

PZT-LTCC monomorph structure was also manufactured in order to test whether

the more precise monolithic process improved the frequency performance of a

wideband harvester. The active material of the harvester was PZ29 (Ferroperm

A/S, Denmark) instead of the PZT-5H used for the PZT-steel harvester. The

passive material was Heraeus Heralock 2000 LTCC tape. The main difference

between the previous prototype and the LTCC embedded prototype was in the

manufacturing of the layered structures. In the previous prototype each layer was

cut with a laser and then glued together. The gluing, which was done by hand,

added errors in the positioning of the layers in the XY-plane and also in the Z-

direction as the glue layer thickness was not precisely controlled. In the LTCC

prototype, however, the positioning was done using a jig and the layers were

sintered together so that a glue layer with arbitrary thickness was not present. The

prototype is shown in Fig. 12.

Fig. 12. The PZT-LTCC harvester without (middle) and with lid (right) [Paper IV,

published by permission of Elsevier Ltd].

The harvester was designed with three beams instead of five as in the PZT-steel

prototype. The beam lengths were calculated using equation (14) and using the

starting length of 18 mm, the overlapping factor, α, of 1 and the damping factor, ξ,

41

of 0.025. Higher overlapping factor for this harvester was chosen as much higher

dimensional accuracy was expected for the monolithic structure. This resulted in a

length series 18, 17.78 and 17.56 mm. Beam width was 3 mm and the layer

thicknesses were 375 and 270 µm for the PZT and LTCC layers, respectively. The

difference between the beam lengths was 220 µm and for the PZT-steel prototype

it was 150 µm. The larger overlapping factor eased the manufacturing of the

harvester since the required length difference was greater than with a smaller

value of α. This is important in structures where the beam lengths are restricted as

in this case where the 25 mm disc diameter restricts the beam length compared to

the previous PZT-steel prototype where a 35 mm disc was used. Smaller disc

diameter was used because of material availability and better manufacturability of

the smaller disc.

4.1.3 Measurement results and comparison between the harvesters

The wide band energy harvesters were measured both as an actuator and a

harvester. The actuator measurements were done using a vibrometer OFV-5000

(Polytech GmbH, Germany) for the displacement measurements. A piezo stack

Piezomechanik Pst 150/7/160 VS12 was used to generate the desired acceleration

for the energy harvesting measurements. The energy harvesting electronics are

shown in Fig. 13 where a common rectifier is used together with a capacitor (1

µF) as the energy storage unite unit.

Fig. 13. The schematic of the energy harvesting electronics [Paper III published by

permission of Suomen Automaatioseura].

42

Actuator measurements were performed in order to assess the resonance

frequencies of the beams. Fig. 14 shows the frequency response curve of the PZT-

steel harvester under 10 Vp-p excitation. It can be observed that beam number 5

was not working, thus making the energy harvester effectively a 4 beam harvester.

Furthermore, the order of the resonances was 1-2-4-3 as opposed to the designed

order of 1-2-3-4. Fortunately this does not weaken the performance since the

spacing of the resonances is still acceptable. Beams number 4 and 5 were located

side by side and since the amplitude of beam number 4 was also attenuated as in

the case of beam 5, it can be said that same problem could be affecting for both

beams. This can be for example the glue layer between the steel and PZT, which

by altering the total thickness locally changes the resonance frequency of the

beams located in the said area. The resonance frequencies for the beams 1, 2, 3

and 4 were 695, 700, 724 and 712 Hz, respectively [Paper III]. According to

equation (12), resonances 695, 703.7, 712.5 and 721.4 Hz were obtained. When

comparing this series with the measured values, errors for the measured and

calculated frequencies were obtained as 0.5 %, 1.6 % and -1.3 %. The first

calculated frequency was assumed to be 695 Hz and the error was therefore 0 %.

As can be seen, the relative errors in the frequencies were small but could still

result in the beams switching places in the frequency domain. In this prototype

the resonance frequencies for beams 3 and 4 were very close to the designed

values as they have switched places, but this was not according to design. The

displacement amplitudes of the resonances were similar for beams 1, 2 and 3, but

beam 4 was attenuated to roughly half of the other beams’ displacement.

43

Fig. 14. The Frequency response of the PZT-steel energy harvester under 10 Vp-p

sinusoidal actuation [Paper III published by permission of Suomen Automaatioseura].

Similar measurements were performed for the PZT-LTCC prototype and the

resulting frequency response curve is shown in Fig. 15. As can be seen, the order

of the resonances was as desired 1-2-3 and the respective resonance frequencies

were 1109, 1137 and 1155 Hz [Paper IV]. The calculated values using equation

(12) were 1100, 1137 and 1165 Hz. Therefore the errors were 0.82, 0 and 0.86 %

for the frequencies. Thus, the errors were essentially half of those of the PZT-

steel prototype. Further insight into the effect of the overlapping factor is gained

when the frequency differences are studied. The differences fres,n-fres,n-1 for the

PZT-steel model were 5, 24, -12 Hz. The calculated values, however, were 8.7,

8.8 and 8.9 Hz. Now the errors for the differences were 74 %, 172 % and 234 %.

These values differ greatly from the absolute errors which were at maximum

1.6 %. The same measured differences for the PZT-LTCC prototype were 28 and

18 Hz and the calculated values were 37 and 28 Hz. Now the errors are 24 and

36 %. As the required frequency difference is larger due to the overlapping factor

α, the relative error also becomes lower. Therefore larger overlapping factors

should be used even though the power output is somewhat reduced because this

ensures that wideband operation is maintained even though errors are introduced

in the manufacturing process. This can however result in bandpass ripple

exceeding the 3 dB level and if that is not acceptable, then lower overlapping

factor should be used. This is a design choice that should be considered when

44

facing high manufacturing tolerances. When comparing the magnitudes of the

resonances between Figs. 14 and 15, it can be noted that for the PZT-LTCC

prototype the magnitudes are in descending order, which is characteristic of beam

resonances in actuation.

Fig. 15. The PZT-LTCC harvester under 1 Vp-p sinusoidal actuation [Paper IV,

published by permission of Elsevier Ltd].

The wideband operation of the harvesters was measured using a frequency sweep

and connecting the harvester into the circuit shown in Fig. 13. In this

measurement, the harvester was attached to a shaker and the frequency of the

shaker was varied while the amplitude remained constant. For the PZT-steel

prototype a 6 µm peak-to-peak displacement was used for the actuation. With

constant displacement, the acceleration amplitude ranged from 36 to 76 m/s2as the

frequency was swept from 550 to 800 Hz. The voltage of the capacitor was

recorded and the energy content of the capacitor was then calculated from the

voltage. The resulting frequency response is shown in Fig. 16. The centre

frequency was 717 Hz, with a 3 dB bandwidth of 54 Hz [Paper III]. The

improvement of the bandwidth when comparing it to a single beam, with the

designed Q-value of 20, is 50 %. Using measured Q-value of 24.6, the

45

improvement was 85 %. For the PZT-LTCC prototype the frequency response is

shown in Fig. 17. In this measurement, the amplitude was 1 µm, as the frequency

was higher than in the PZT-steel case. This resulted in accelerations between 20

and 33 m/s2. The centre frequency was 1147 Hz and the bandwidth was 62 Hz

[Paper IV]. Therefore the actual improvement in the bandwidth compared to a

single beam was 5.4 times larger i.e. 540 % since the single beam’s measured

bandwidth was 11 Hz. The measured Q-value of the single beam was 103 on

average and the value used in the design phase was 20. In comparison, the PZT-

steel prototype had an average Q-value of 24.6 which deviated only 23 % from

the value used in the calculations. The high Q-value of the PZT-LTCC prototype

is also seen in the Fig. 17 as the pass-band of the harvester had a considerable

ripple which should not exist with such a low overlapping factor. For the PZT-

steel prototype the pass-band notch was not due to a high Q-value, because the Q-

value was close to that of the value used in the calculations, but rather due to the

low output of beam number 4, evident in Fig. 14.

Fig. 16. The buildup voltage and energy of the capacitor on PZT-steel

energy harvester [Paper III published by permission of Suomen Automaatioseura].

46

Fig. 17. The buildup voltage of the capacitor of the PZT-LTCC harvester, the voltage

across the optimal load resistance and the energy of the capacitor.

As the PZT-LTCC prototype showed, the Q-value can differ greatly from design

to design. This affects the resonance frequency placement and if the error in the

Q-value estimation is large, the pass-band of the energy harvester will generate

ripple. Another important design decision is the choice of manufacturing process.

As was shown, the less precise manufacturing process where layers were glued by

hand in the PZT-steel prototype led to large errors in the resonance frequencies of

the beams. In contrast, the LTCC process proved to be very good in

manufacturing precise geometries and therefore the resonance frequencies of the

beams were very close to the designed values. The third important aspect is the

choice of the overlapping factor as it can to some extent alleviate the tolerances in

manufacturing, especially when using a large number of beams. This is of course

at the expense of bandwidth and energy, since a higher overlapping factor

increases the bandwidth but decreases the power and by using a lower

overlapping factor the effect is the opposite. The error analysis showed the high

sensitivity of length variations to the resonance frequency, and this can be used as

a design tool in the selection of manufacturing process and overlapping factor.

The performance of the harvesters could be improved by introducing mass at the

tip of the beams, because that lowers the resonance frequencies and also improves

the power output of the energy harvester [86]. In this work, masses were not

47

introduced, because the focus was on the frequency tuning by length variation of

beams and other variables were kept to a minimum.

The small signal analysis proved to be a useful tool when designing wideband

multibeam harvesters, since together with simple beam equations, process

tolerances and material Q-value information it could be used in designing the

beam resonance frequencies which fulfilled the specified frequency range

properties. In [26] a similar laser processed structure was developed and the

claimed improvement of the bandwidth was 200 % compared to that of a single

beam for a 6 beam structure and 33 % per beam. The results were 50 % and

540 % for the PZT-steel harvester and PZT-LTCC harvester, respectively.

Improvements per beam were 12.5 % and 180 %. A MEMS implementation in

[77] claimed resonance frequencies between 237 – 244.5 Hz with four beams,

which roughly translates to a relative bandwidth of 3.12 %, whereas the PZT-steel

prototype and the PZT-LTCC prototype obtained 7.5 % and 5.4 %, respectively.

Besides performance enhancements, the PZT-LTCC prototype provides an

integrated package for the harvester and also enables the integration of

harvesting, and sensor electronics into the harvester itself. This then enables very

robust harvesters for environmentally demanding applications, since LTCC is

both a circuit board and a package at the same time.

4.2 Modelling energy harvesters for walking

Wearable electronics have been developed to be used, for example, in firefighters’

outfits and in military attires. Various sensors embedded in shoes, gloves and

other wearable items need power, and batteries or accumulators are usually used.

However, the problem with these is the weight and the need to recharge or replace

them. Energy harvesting is a natural solution for wearable electronics. A wide

range of solutions exist for generating power from the movement of the body

such as harvesting the heel strike of the shoe [87], the pressure of the foot [88],

the movement of the knee joint [89], the straps of the backpack [90] and the heat

from the body[91]. In this work, however, the pressure and heel strike of the foot

were used as the energy source. Energy harvesting from walking presents many

challenges. The low operation frequency of ~ 1 Hz limits the use of resonating

structures and therefore the coupling of energy from the mechanical domain into

the electrical domain in piezoelectric materials. Also, the harvester should not

hinder the movement or provide discomfort to the user of the device. The energy

density of the harvester device should also be maximized since the space

48

available is limited. Furthermore, when embedding energy harvesters into shoes,

the force experienced by the harvester is not a simple step function, but rather has

a defined shape, which affects the power output and also the FEM modelling of

the device.

In this chapter, the performance of a cymbal harvester is studied using FEM-

modelling. The mechanical input into the harvester is measured and the results are

used as an input to the FEM simulator. The mechanical input in this case is the

force experienced by the harvester when mounted inside the sole of a shoe during

walking. FEM modelling provides a tool for understanding the behaviour of the

piezo electric harvester under high stresses during walking. The harvester in

question is a cymbal type transducer, which is typically used as an actuator [92-

93]. FEM modelling is a commonly used tool in cymbal [93-98], and also in

harvester design. Sinusoidal excitation was used in [99-100] with a relatively high

excitation frequency of 100 Hz. The power output of simple piezoelectric discs

was studied in [101]. FEM and analytical methods was used in [102], to study the

effect of an additional steel layer attached to the piezoelectric layer in a cymbal

configuration. In [103], circumferential slots were incorporated into the endcaps

of the cymbal and FEM together with analytical methods was used as a tool.

However, the accurate simulation of cymbals subjected to the actual walk profile

has not previously been studied. The incorporation of arbitrary waveforms,

nonlinear mechanics and large displacements into a single cymbal energy

harvester simulation is a novel approach which gives valuable information about

cymbal energy harvesters under high load conditions.

4.2.1 The design of the shoe mounted energy harvester

The shoe mounted piezoelectric harvester was designed to be embedded into the

sole of the shoe. Furthermore, the purpose of the study was to create a FEM-

model of the cymbal structure and to model the energy generation from walking.

Since the force experienced by the harvester is not sinusoidal, a transient model

with a relevant force profile is needed. Also, energy harvesting simulations

require the combination of both the mechanical and electrical domain simulations.

This model addresses all these requirements. The basis of the cymbal structure

was a PZT-5H disc 35 mm in diameter into which brass connector rings were

glued together with convex steel endcaps. The endcaps were pressed into shape

using a mould. This process was done manually and therefore the endcap shape

was not entirely uniform. The cymbal and its dimensions are shown in Fig. 18.

49

Fig. 18. The cymbal structure and the individual parts of the shoe mounted

piezoelectric harvester [Paper V, published by permission of SAGE publishing].

4.2.2 The FEM model of the energy harvester

The profile of the endcap was measured using a micrometer screw gauge and one

half of the profile was used in the model by mirroring it into the axial geometry.

This introduced some errors into the model since the endcap geometry was not

modelled in all quarters of the actual geometry. The measured profile is shown in

Fig. 19.

Fig. 19. The measured profile of the endcap used in model and measurements

[Paper V, published by permission of SAGE publishing].

50

The geometry of the model was designed to be parametric where all of the

geometric properties were added as variables that could be modified during

simulation passes. The shape of the endcap was modelled as a normalized shape

vector ranging from 0 to 1. The concavity and radius together with other

geometric parameters defined the actual points of the endcap. This design

provided a powerful tool for optimizing cymbal designs as in [22]. The meshed

geometry of the axial-symmetric design is shown in Fig. 20 together with plastic

cushions at the top and bottom of the cymbal. The plastic cushions were

incorporated into the model because the cymbal was installed into a plastic holder

which was then installed into a shoe sole. The force of the heel pressure was

therefore conveyed into the cymbal through these plastic cushions in a controlled

manner. The measurement system is presented in detail in Paper V.

The material parameters for the piezoelectric material, steel and the plastic

cushions are shown in Table 2. These values were obtained from the

manufacturer’s catalog [85] for the PZT. The steel’s and nylon plastic cushion’s

values were obtained directly from the Comsol software (Comsol Multiphysics

4.2 a). As can be seen from Fig. 20, triangular elements were used. The order of

the elements was quadratic and therefore the accuracy of the simulations was

adequate compared, for example, to constant strain triangle (CST) elements which

cannot properly calculate the stresses inside the triangle. Furthermore, the

geometric nonlinearities setting was applied for the simulations. This setting is

also known as the large deformation setting in some other FEM softwares. This is

necessary because the deformation levels are large in the simulations; the applied

stroke for the cymbal in the simulations was 1.5 mm, which is near the point

where the endcap touches the piezo disc.

Fig. 20. The meshed axial-symmetric geometry of the cymbal [Paper V, published by

permission of SAGE publishing].

51

The lower plastic pad was fixed from the bottom as a fixed boundary condition

and the top pad was in turn actuated with the input force. The force experienced

by the harvester was measured from the prototype during actual walking. This

force waveform was then used as an input for the top pad as a force type

boundary condition in the transient simulations. The force measurement data was

filtered to remove high frequency content in order to increase the step size in

simulations and therefore shorten the total simulation time. This filtered force

profile is shown in Fig. 21. As the results show, the pace of walking was roughly

1 Hz. This is also shown by [104-106] to be a good average pace for walking.

Fig. 21. The force input for the shoe mounted energy harvester at 0.5 mm stroke

[Paper V, published by permission of SAGE publishing].

The electrical boundary conditions were set as ground for the bottom electrode of

the PZT disc and the top electrode was set as a terminal connection for an

external SPICE circuit which consisted of a simple load resistor. This external

circuit was simulated concurrently with the FEM simulations using the same

Comsol software. The SPICE simulations provided an easy method for

calculating the power dissipated by the external load resistor. Furthermore, the

optimal load resistance was easily obtained using parametric simulations where

the resistance was swept.

52

4.2.3 The simulation results of the energy harvester

The first simulation was carried out as a static displacement vs. force simulation

where the stroke was applied to the top pad of the energy harvester and the

resulting force on the same surface was recorded, as shown in Fig. 22. The same

curve was also generated by measuring the prototype and these results are also

shown in the figure. As can be seen, the simulated values largely agreed with the

measured values although some differences appeared. Namely, in the area

between 0.6 to 1.6 mm the simulated forces were below the measured values and

also the nonlinearity was greater in the simulated curve than in the measured one.

The maximum error occurred at 1.1 mm with 8.8 % error. It has to be noted also

that the most significant contributors to this displacement vs. force curve were the

endcaps of the component, because the material of the piezoelectric disc deforms

very little compared to that of the endcaps. Therefore the mechanical properties of

the endcaps will have the most effect on the total performance of the harvester, as

will be shown later. The simulation type can also explain the difference between

measured and simulated results, since the model of the endcap was only one half

of an endcap mirrored to generate an axial symmetric geometry. From Fig. 22, the

average spring constants for the measured and simulated cases were obtained as

34.1 kN/m and 30.4 kN/m, respectively. As the nonlinearity of the curve is quite

strong a more general nonlinear temporal spring constant can be obtained by first

fitting a curve for the force vs. displacement curve. The force in the case of the

simulated model is

2.014 ∙ 10 4.114 ∙ 10 4.0808 ∙ 10 1.497 (20)

, where d is the displacement of the endcaps and the F is the reactive force on the

pad. The temporal spring constant is by definition the derivative of F with respect

to d as

6.042 ∙ 10 8.228 ∙ 10 4.0808 ∙ 10 (21)

This function reaches its minimum of 12.8 kN/m at 0.68 mm and its maximum of

63.7 kN/m at 1.6 mm when the displacement is restricted to 0.4 to 1.6 mm as in

the measurements. As Fig. 23 shows, the spring constant is highly nonlinear,

especially in the modelled case.

53

Fig. 22. The displacement vs. force of the cymbal actuator [Paper V, published by

permission of SAGE publishing].

For the measured harvester the force as a function of displacement was

1.137 ∙ 10 2.122 ∙ 10 3.955 ∙ 10 1.194 (22)

and the spring constant

3.411 ∙ 10 4.244 ∙ 10 3.955 ∙ 10 (23)

The minimum spring constant was located at 0.62 mm with a value of 26.4 kN/m.

The maximum at 1.6 mm resulted in 59 kN/m. When comparing the simulated

and measured spring constants, the measured spring constant had a higher

minimum and slightly lower maximum, therefore the simulated spring constant

exhibited a higher nonlinearity compared to the measured case. The minimum

occurred roughly at the same displacement. Furthermore, when comparing the

maximum deviation from the average spring constant obtained from the linear

fitting, the simulated case deviated 33.3 kN/m and the measured case deviated

24.9 kN/m. This shows that the average spring constant cannot be used in cases

where large deformations are encountered such as in a cymbal under high loads.

The spring constants in equations (21) and (23) are plotted in Fig. 23.

54

Fig. 23. The spring constant versus displacement of the measured and simulated

end cap of the shoe mounted piezoelectric harvester.

The force applied to the top of the end cap is transformed by the end cap into

radial stress in the piezoelectric disc. This transformation was simulated with a

similar simulation as in Fig. 22, where the force versus displacement was

modelled. In this case, however, the force was used as an input and the

displacement of the end cap and the von Mises stress inside the piezoelectric disc

was simulated. These results are shown in Fig. 24. As can be seen, at 30 N the

stress induced into the piezo disc started to drop. When comparing figures 23 and

24, it can be seen that the spring constant started to rise after 0.8 mm and at the

same displacement the change in stress growth in Fig. 24 started to decrease.

Therefore the drop in stress relaying properties of the end cap can indeed be

attributed to the nonlinear spring constant. Deformations of the end cap under

different loads are shown in Fig. 25. As can be seen, the deformations are

noticeable and at 30 N a recess had formed in the middle of the end cap. This

deformation therefore stiffened the structure giving rise to the highly nonlinear

spring constant shown in Fig. 23.

55

Fig. 24. Displacement and piezo stress as a function of force of the cymbal harvester

[Paper V, published by permission of SAGE publishing].

Fig. 25. The end cap shape under different loads [Paper V, published by permission of

SAGE publishing].

The power generation simulations were done using the force profile in Fig. 21.

The simulation was carried out as a transient simulation with the force applied to

56

the top pad. A similar sine signal was also used for comparison, with a peak-to-

peak value adjusted to match the walk profile. The results are shown in Fig. 26.

Fig. 26. The power generation with the walk and sine force profile under 17 N peak

load with optimal 2.63 MΩ load resistance [Paper V, published by permission of SAGE

publishing].

The load resistance was adjusted to 2.63 MΩ which was measured to be the

optimal resistance [Paper V]. The optimal resistance was also simulated using

transient simulation with the walk profile. In that case the optimal resistance was

3.0 MΩ which is very close to the measured value and therefore the use of the

measured value is also justified in the simulations. These results are shown in Fig.

27. As can be seen in Fig. 26, the walk profile produced a large peak when the

heel compressed the cymbal. When the heel was in the static pressure mode as

seen in Figs. 21 and 26 during 0.2 to 0.4 seconds, the power decays to zero and

when the heel rises and the cymbal experiences decompression the power rises

again to produce a similar peak but only smaller. When comparing the walk

profile with the sine profile, it can be seen that the compression cycle produced

roughly a 4.5 times higher peak compared to the sine profile [Paper V]. The

decompression cycle, however, produced a similar peak to the sine peak. The

compression cycle’s rise time was 60 ms and in the decompression cycle it was

184 ms. The shorter rise time explains the higher power output during the

compression cycle and also why the sine signal produced such a low power

57

output. The difference in power levels is also seen in Fig. 27, where the walk

profile produced ~100 µW more power in the 2.63 MΩ load resistor, which is

167 % higher power generation compared to sine signal. This result emphasizes

the importance of correct input signal modelling in energy harvesting simulations

where the input signal deviates from a sine signal.

Fig. 27. The power vs load resistance of the walk and sine input [Paper V, published by

permission of SAGE publishing].

The average power generations were measured using 12 different stroke values

ranging from 0.5 mm to 1.6 mm. The same displacements were also used in

simulations. In these simulations the profile in Fig. 21 was scaled to match the

force profile at the given stroke. The load resistance was adjusted to 2.63 MΩ in

both the measurements and simulations. Both results are shown in Fig. 28, where

the average power versus stroke distance is shown. The simulations were carried

out as transient simulations as in the case of the simulations depicted in Fig. 27

and the average power was calculated from one walk cycle and the result was

plotted. As can be seen from Fig. 28, the simulated values agree well with the

measured values up to 1.2 mm displacements. At 1 mm displacement, the

measurements show a deviation, which is probably caused by a measurement

error. The power output for the simulated model peaked at 1.3 mm with 640 µW.

The measured power peaked at 1.5 mm with 780 µW. The error between the

58

simulated and measured power below 1.3 mm was 7.1 % with the exception of

1 mm displacement where the error was 14.7 %. Above 1.3 mm the simulated and

measured values diverged and the simulated values saturated. As discussed

earlier, the saturation derives from the nonlinear spring constant of the end caps.

The same saturation did not occur until 1.5 mm displacement in the case of the

measured values. Fig. 23 shows that the measured spring constant exhibited

considerably greater linearity compared to the simulated case. This can also

explain the different saturation behaviour of the measured cymbal compared to

the simulated model. Therefore, when generating a model for a cymbal energy

harvester, a comparison between the spring constants is encouraged in order to

gain insight into establishing reliable models for energy harvesting simulations

with high deformations.

As the results showed, the developed model accurately modelled the

behaviour of the cymbal piezoelectric harvester. The accurate input signal and

geometry modelling, together with the combination of mechanical and electrical

simulations, generated a useful tool for cymbal type energy harvester design. This

model can be extrapolated into other topologies and operational environments and

thus provides added value to large stroke energy harvester designs.

Fig. 28. The average measured and simulated power versus displacement of the

cymbal harvester [Paper V, published by permission of SAGE publishing].

59

5 Conclusions

The main scope of this thesis was to develop methods to combine the equivalent

circuit analysis and finite element methods to improve the design process of

actuator and energy harvester dynamics. In addition, a new FPI structure was

developed to provide more cost effective alternative to the state of the art FPI

actuator. FEM analysis also provided further insight of the cymbal energy

harvester under large displacements during walking. The small signal parameters

were obtained either from measurements or from the FEM model. The small

signal model of the component could then be used in estimating resonance

frequencies of beams either in FPI actuators or wideband energy harvesters. In

addition, the nonlinear spring constant of a cymbal energy harvester was

discussed and its effect on the saturation behaviour was investigated in relation to

stroke displacement or input force.

The Butterworth-Van-Dyke equivalent circuit was developed for a pre-

stressed piezoelectric actuator. The small signal parameters were obtained from a

series of measurements. The model was tested against the component with

resonance measurements with varied added masses. The model deviated only

2.5 % at maximum. As the equivalent circuit reveals, the series capacitance can be

used in tuning the resonance frequency of the actuator. This was tested both with

the actual component and the small signal model and the calculated results

deviated only 14 % from the measured values. It was also noted, that by adding

a series capacitor into the piezoelectric component, the internal capacitance is

decoupled from the input/output impedance of the external electronics. This

provides a method to tune the resonance frequency of the energy harvester by

electrical means in low load impedance situations.

An FPI actuator was designed and simulated with PZ-29 as the active

material and LTCC as the passive material. The design featured a flexural hinged

mirror holder which alleviated stress in the mirror. This stress can bend the mirror

and therefore degrade FPI performance by affecting the local gap length. The

stress levels were also studied in the hinge area made of LTCC material and they

were confirmed to be well below the fracture limit. The reduction of stress levels

induced into the mirror enables the use of thinner and lighter mirrors which then

improves the transient properties of the FPI-actuator. In this work, the dynamical

parameters, namely the spring constant, k, and the effective mass, meff, of the

actuator were obtained using simulations. Eigenfrequency simulations were

carried out with a series of added masses. Resulting resonance frequencies were

60

then used in calculating the effective mass, meff , and the spring constant. The

small signal model was then compared to measurements where a mass was added

to the mirror holder. The resonance frequencies between the model and the actual

component differed only 0.8 % when mirror was attached, thus validating the

small signal approach. The results also showed that a small TO-3 sized FPI-

actuator can be made with beam structure instead of the state-of-the-art stack

actuators.

Two different types of wideband energy harvesters were designed to collect

energy from a vibrating environment and one cymbal type energy harvester to

collect energy from walking. A conventional PZT-Steel energy harvester was

developed and characterized. This harvester contained five differently tuned

beams, with the purpose of generating a series of resonances that are tuned in a

controlled manner to provide the wideband operation. The resonance frequencies

were tuned by controlling the beam length and thus the effective mass and the

spring constant of the beam. This approach proved effective as the difference

between the measured and calculated resonance frequencies was 1.6 % at

maximum, although two beams had switched places when put into order of their

resonances.

The second wideband harvester was a PZT-LTCC energy harvester which had

only three beams. This manufacturing process was more controlled because the

layers were stacked using a jig instead of manual alignment as in the PZT-Steel

case. The beams were tuned similarly to the PZT-Steel case apart from the

overlapping factor, α, which was 1 instead of 0.5 for the PZT-Steel case. The

results showed that the resonance frequencies differed only 0.86 % from the

calculated values which is nearly two times better than the PZT-Steel case. The

resulting frequency response of the energy harvester, however, exhibited ripple

which was an indicator of an incorrect damping factor, i.e. Q-value, assumption.

Surprisingly, the actual Q-value of the beam was five times that of the designed

value. This manifested itself as a pass-band ripple for the energy harvester.

Nevertheless, the PZT-Steel prototype did not suffer from this, since the designed

and measured Q-values were close. This proved that the overlapping factor and

Q-value have to be carefully decided and measured in the design process. In

addition, the manufacturing process has a very large influence on the resonance

frequencies, because the tolerances of the process can affect the geometry of the

component and therefore the resonance frequency. This was shown by an error

analysis, where the error on the beam length was studied. A method was

developed to incorporate this error into the overlapping parameter selection.

61

Small signal modelling when used in conjunction with correct Q-values, can

be used to tailor the bandwidth of multibeam harvesters. The design parameters

have to be selected according to manufacturing tolerances to optimize the

bandwidth. This was demonstrated using two different manufacturing processes

and the results gave valuable information about the tailoring of the design

parameters.

The energy harvester for walking was developed using a cymbal harvester.

The shape of the endcaps was measured and these results were then input into the

FEM model. As the harvester was designed to be embedded into the sole of a

shoe, the actual force profile of walking was measured and again inputted into the

model. This enabled FEM simulations with a high accuracy in simulating the

power levels obtained from walking. The almost completely parameterized model

provides a powerful tool for further optimizing cymbal type energy harvesters.

The model was first verified by carrying out a force vs. displacement simulation

as well as a similar measurement with the actual prototype. The results differed

by only 8.8 %, which was caused largely by the difference in the endcaps’ spring

constant. The spring constants were compared between the simulated and the

measured cases and the simulation model exhibited larger nonlinearity compared

to the measured component. This proved that the endcap modelling differed from

the actual component. However, as the power generation simulations and

measurements were carried out, the simulated model followed closely the values

of the measured component up to 1.3 mm displacement. After that the model

started to saturate. The measured component exhibited similar saturation, but at

larger displacements. Again, the stiffening of the structure was found to be the

cause of this behaviour. The endcaps’ ability to transfer input force into radial

stress in the piezoelectric disc showed a maximum at around 30 N after which the

stress started to drop and therefore the output energy of the piezoelectric

component saturated. This was a result of the stiffening of the endcaps under high

loads. The presented simulation tool provides a powerful method for optimizing

the endcap geometry to combat the stiffening encountered in this work.

This work presented methods to combine finite element techniques and small

signal analysis to improve the design process of actuator and energy harvester

dynamics. The small signal parameter, k, was also used in explaining the

saturation behaviour of the cymbal energy harvester, where it was found to be

nonlinear. This work presented novel methods to tailor the bandwidth of

wideband energy harvesters according to manufacturing tolerances and material

properties, namely Q-factor. The simulation model of a cymbal type energy

62

harvester used in a high stroke energy harvesting environment is a novel

contribution to energy harvesting. This work also proved that the accurate

modelling of the input signal together with time based transient modelling is an

essential part of high stroke simulations. The generated model is also expandable

to other types of energy harvesters and the methods developed can be used in a

variety of different energy harvesting simulations and harvester development.

63

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Original papers

I Leinonen M, Juuti J & Jantunen H (2005) Equivalent circuit based modification of a pre-stressed piezo actuator. Proc. 4th Int. Conf. on Smart Systems, Seinäjoki.

II Sobocinski M, Leinonen M, Juuti J & Jantunen H (2012) A Piezoelectric Active Mirror Suspension System Embedded Into Low-Temperature Cofired Ceramic. IEEE trans. on Ultrasonics, Ferroelectrics and frequency control 59(9): 1990–1995.

III Leinonen M, Palosaari J, Juuti J & Jantunen H (2011) Piezoelectric energy harvester for vibrating environments using multiple beam topology for wideband operation. AP XIX Proceedings 41, Helsinki.

IV Sobocinski M, Leinonen M, Juuti J & Jantunen H (2011) Monomorph piezoelectric wideband energy harvester integrated into LTCC. Journal Of European Ceramic Society 31(5): 789–794.

V Leinonen M, Palosaari J, Juuti J & Jantunen H (2014) Combined electrical and electromechanical simulations of a piezoelectric Cymbal harvester for energy harvesting from walking. Journal of Intelligent Material Systems and Structures, 25(4): 391–400.

Reprinted with permission from Frami OY (Paper I), IEEE (Paper II), Suomen

automaatioseura (Paper III), Elsevier Ltd (Paper IV) and Sage publishing (V).

Original publications are not included in the electronic version of the dissertation.

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